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Introduction to Integers
Introduction to Numbers
Natural Numbers : The collection of all the counting numbers is called set of natural numbers. It is denoted by N = {1,2,3,4….}
Whole Numbers: The collection of natural numbers along with zero is called a set of whole numbers. It is denoted by W = { 0, 1, 2, 3, 4, 5, … }
Properties of Addition and Subtraction of Integers
Closure under Addition and subtraction
For every integer a and b, a+b and a–b are integers.
Commutativity Property for addition
for every integer a and b, a+b=b+a
Associativity Property for addition
for every integer a,b and c, (a+b)+c=a+(b+c)
Additive Identity & Additive Inverse
Additive Identity
For every integer a, a+0=0+a=a here 0 is Additive Identity, since adding 0 to a number leaves it unchanged.
Example : For an integer 2, 2+0 = 0+2 = 2.
Additive inverse
For every integer a, a+(−a)=0 Here, −a is additive inverse of a and a is the additive inverse of-a.
Example : For an integer 2, (– 2) is additive inverse and for (– 2), additive inverse is 2. [Since + 2 – 2 = 0]
Properties of Multiplication of Integers
Properties of Multiplication of Integers
Closure under Multiplication
For every integer a and b, a×b=Integer
Commutative Property of Multiplication
For every integer a and b, a×b=b×a
Multiplication by Zero
For every integer a, a×0=0×a=0
Multiplicative Identity
For every integer a, a×1=1×a=a. Here 1 is the multiplicative identity for integers.
Associative property of Multiplication
For every integer a, b and c, (a×b)×c=a×(b×c)
Distributive Property of Integers
Under addition and multiplication, integers show the distributive property.
i.e., For every integer a, b and c, a×(b+c)=a×b+a×c
These properties make calculations easier.
Division of Integers
Division of Integers
When a positive integer is divided by a positive integer, the quotient obtained is a positive integer.
Example: (+6) ÷ (+3) = +2
When a negative integer is divided by a negative integer, the quotient obtained is a positive integer.
Example: (-6) ÷ (-3) = +2
When a positive integer is divided by a negative integer or negative integer is divided by a positive integer, the quotient obtained is a negative integer.
Example: (-6) ÷ (+3) =−2 and Example: (+6) ÷ (-3) = −2
The Number Line
Number Line
Representation of integers on a number line
On a number line when we
(i) add a positive integer for a given integer, we move to the right.
Example : When we add +2 to +3, move 2 places from +3 towards right to get +5
(ii) add a negative integer for a given integer, we move to the left.
Example : When we add -2 to +3, move 2 places from +3 towards left to get +1
(iii) subtract a positive integer from a given integer, we move to the left.
Example: When we subtract +2 from -3, move 2 places from -3 towards left to get -5
(iv) subtract a negative integer from a given integer, we move to the right
Example: When we subtract -2 from -3, move 2 places from -3 towards right to get 1
Addition and Subtraction of Integers
The absolute value of +7 (a positive integer) is 7
The absolute value of -7 (negative integer) is 7 (its corresponding positive integer)
Addition of two positive integers gives a positive integer.
Example : (+3)+(+4) = +7
Addition of two negative integers gives a negative integer.
Example : (−3)+(−4) = −3−4=−7
When one positive and one negative integers are added, we take their difference and place the sign of the bigger integer.
Example : (−7)+(2) = −5
For subtraction, we add the additive inverse of the integer that is being subtracted, to the other integer.
Example : 56–(–73) = 56+73 = 129
Introduction to Zero
Integers
Integers are the collection of numbers which is formed by whole numbers and their negatives.
The set of Integers is denoted by Z or I. I = { …, -4, -3, -2, -1, 0, 1, 2, 3, 4,… }
Properties of Division of Integers
Properties of Division of Integers
For every integer a,
(a) a÷0 is not defined
(b) a÷1 = a
Note: Integers are not closed under division
Example: (– 9) ÷ (– 3) = 2. Result is an integer.
and (−3)÷(−9)= 1/3. Result is not an integer.
Multiplication of Integers
Multiplication of Integers
Product of two positive integers is a positive integer.
Example : (+2)×(+3) = +6
Product of two negative integers is a positive integer.
Example :(−2)×(−3) = +6
Product of a positive and a negative integer is a negative integer.
Example :(+2)×(−3) = −6 and (−2)×(+3) = −6
Product of even number of negative integers is positive and product of odd number of negative integers is negative.
These properties make calculations easier.